Yes...but they will ....if they can furnish and do what he’s asked, according to this posted here.
This looks to me like this is mostly for those who try to enter illegally...those who go through port of entry will be heard to determine if or not asylum is granted.
....many of these people have joined together calling themselves all one big family.....one woman bragged about all those she’s ‘adopted’ with the intention of saying they are all of the save last name when they arrive. Now I doubt they can pull it off but this just shows they will say whatever will get them in.....even if it won’t.
But asylum must be accepted at the first country that offers it. And Mexico offered it to them all.
Trump doesn’t mention that, as it is a detail. But it is there and will prevent asylum claims from being accepted.